Dirichlet dynamical zeta function for billiard flow

Abstract

We study the Dirichlet dynamical zeta function $\eta_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large ${\rm Re}\: s$ we have $\eta_D(s) =\sum_{n= 1}^{\infty} a_n e^{-\lambda_n s}, \: a_n \in \mathbb R$ and $\eta_D$ admits a meromorphic continuation to $\mathbb C$. We obtain some conditions of the frequencies $\lambda_n$ and some sums of coefficients $a_n$ which imply that $\eta_D$ cannot be prolonged as entire function.